To solve this problem, let’s analyze the given equation and the required expression.

1. Given Equation: $x^2 + y^2 - 2x = 2024$

   Rewrite it by completing the square on $x$:

   $x^2 - 2x + y^2 = 2024$ $(x - 1)^2 - 1 + y^2 = 2024$ $(x - 1)^2 + y^2 = 2025$

   This represents a circle with center at $(1, 0)$ and radius $45$ (since $2025 = 45^2$).

2. Objective: We need to find the maximum possible value of $(x - 6)^2 + (y - 12)^2$.

   This expression represents the square of the distance from any point $(x, y)$ on the circle to the point $(6, 12)$.

3. Approach: The maximum distance from any point on the circle to $(6, 12)$ will occur when the point on the circle is directly opposite $(6, 12)$ along the line joining $(1, 0)$ and $(6, 12)$.

4. Distance Calculation:

   • Calculate the distance from the center of the circle, $(1, 0)$, to the point $(6, 12)$: $d = \sqrt{(6 - 1)^2 + (12 - 0)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
   • The maximum distance from $(6, 12)$ to a point on the circle will be the radius $45$ added to this distance: $\text{Maximum distance} = 13 + 45 = 58$

5. Final Answer: The square of this maximum distance is: $58^2 = 3364$

Answer: The maximum possible value of $(x - 6)^2 + (y - 12)^2$ is $3364$.

Would you like a detailed breakdown or have any questions?

Related Questions

1. How do you complete the square for a general quadratic expression?
2. Why does maximizing the distance to $(6, 12)$ require finding the farthest point on the circle?
3. What are the properties of a circle that help in solving distance-based maximization problems?
4. Can we use similar methods to find minimum distances from points outside a circle?
5. How would the solution change if the radius or center of the circle were different?

Tip

When working with geometric optimization problems, always try visualizing the setup to understand the points of maximum and minimum distances.